Principal component analysis (PCA) is a well-studied multivariate embedding technique for multivariate data with a wide array of applications, including topic modeling, micro-array analysis, and computer vision (for a recent tutorial, see [1]}). The objective of multivariate data embedding, sometimes referred to as multidimensional scaling, is to reduce the dimension of an \(n \times p\)  data matrix \(\mathbf {X}\)  whose rows are assumed to be \(n\)  independent observations from the population \(\mathcal {X} \in \mathbf {R}^p\) . Applying PCA to a (tabular) matrix \(\mathbf {X}\)  results in two primary data objects of interest. The first is a collection of \(p\)  orthonormal vectors of length \(p\) , called the principal components (PCs), where the \(j\) th vector is the linear combination of the columns of \(\mathbf {X}\)  that explain the \(j\) th most variability in \(\mathbf {X}\) . The second output of PCA is a collection of \(p\)  scalars, known as the loadings, whose \(j\) th value quantifies the variability of \(\mathbf {X}\)  explained by the \(j\) th PC. The PCAN and sPCAN methods are generalizations of PCA developed for the analysis of network samples. Both methods benefit from the advantages of PCA for exploration, visualization, and comparative analysis of network samples. We note that there is a growing literature on the development and analysis of multivariate PCA for high-dimensional (tabular) data which may provide fruitful future directions in the analysis and use of PCA for network-valued data with special high-dimensional structure (see for example [2]}, [3]}).
